Let ( A B 0 ) ( 0 A B ) ( B 0 A ) be a 3Nx3N real matrix with A,B,0 each NxN and 0 an all-zeros matrix.

What is the necessary and sufficient condition for that matrix to be orthogonal, i.e. that its transpose also be its inverse?

This problem statement can be considered ambiguous.*But if you derive a good parametric form for (A B) you will know it.*

(I already "know the answer." I post from curiosity: Is this a VERY easy problem, or just an easy problem.)

The problem does have some historic interest. Such transforms were in vogue, briefly(?), in the 1990's for signal compression. A certainComputer Sci. Professor published several papers and even a full-length textbook on Nx2N lapped orthogonal transforms, with numerous Remarks and Theorems which would all have been trivial Corollaries had he solved the problem posed above.